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Adaptive robust control of nonholonomic systems with stochastic disturbances. (English) Zbl 1117.93027
Summary: This paper deals with nonholonomic systems in chained form with unknown covariance stochastic disturbances. The objective is to design the almost global adaptive asymptotical controllers in probability u 0 and u 1 for the systems by using discontinuous control. A switching control law u 0 is designed to almost globally asymptotically stabilize the state x 0 in both the singular x 0 (t 0 )=0 case and the non-singular x 0 (t 0 )0 case. Then the state scaling technique is introduced for the discontinuous feedback into the (x 1 ,x 2 ,,x n )-subsystem. Thereby, by using backstepping technique the global adaptive asymptotical control law u 1 has been presented for (x 1 ,x 2 ,,x n )-subsystem for both different u 0 in non-singular x 0 (t 0 )0 case and the singular case x 0 (t 0 )=0. The control algorithm validity is proved by simulation.
MSC:
93B35Sensitivity (robustness) of control systems
93C42Fuzzy control systems
93E03General theory of stochastic systems
93E15Stochastic stability
References:
[1]Brockett, R. W., Asymptotic stability and feedback stabilization, in Differential Geometric Control Theory (eds. Brockett, R. W., Millman, R. S., Sussmann, H. J.), Boston: Birkhauser, 1983, 181–191.
[2]Astolfi, A., Iscontinuous control of nonholonomic systems, Systems and Control Letters, 1996, 27: 37–45. · Zbl 0877.93107 · doi:10.1016/0167-6911(95)00041-0
[3]Astolfi, A., Schaufelberger, W., State and output feedback stabilization of multiple chained systems with discontinuous control, in Proceeding of 35th IEEE Conference on Decision and Control, Kobe, 1996, 1443–1447.
[4]Jiang, Z. P., Iterative design of time-varing stabilizers for multi-input systems in chained form, System and Control Letters, 1996, 28: 255–262. · Zbl 0866.93084 · doi:10.1016/0167-6911(96)00029-1
[5]Murray, R., Sastry, S., Nonholonomic motion planning: steering using sinusoids, IEEE Trans. Automatic Control, 1993, 38: 700–716. · Zbl 0800.93840 · doi:10.1109/9.277235
[6]Morin, P., Pomet, J., Samson, C., Developments in time-varying feedback stabilization of nonlinear systems, Preprints of Nonlinear control systems design symposium (NOLCOS’98), Enschede, 1998, 587–594.
[7]Arnold, V. I., Novikov, S.P. (eds.), Dynamical Systems VII, Berlin: Springer-Verlag, 1994.
[8]Bennani, M. K., Rouchon, P., Robust stabilization of flat and chained systems, European Control Conference, 1995.
[9]Hespanha, J. P., Liberzon, S., Morse, A. S., Towards the Supervisory Control of Uncertain Nonholonomic Systems, Preceedings of the American Control Conference, 1999, 3520–3524.
[10]Wang, Z. P., Ge, S. S., Lee, T. H., Robust adaptive neural network control of uncertain nonholonomic systems with strong nonlinear drifts, IEEE Transactions on Systems, Man, and Cybernetics, 2004, 34(5): 2048–2059. · doi:10.1109/TSMCB.2004.833340
[11]Laiou, M. C., Astolfi, A., Discontinuous control of high-order generalized chained systems, Systems and Control Letters, 1999, 37: 309–322. · Zbl 0948.93042 · doi:10.1016/S0167-6911(99)00037-7
[12]Kushner, H. J., Stochastic Stability and Control, New York: Academic, 1967.
[13]Isidori, A., Nonlinear Control Systems, 3rd edition, New York: Springer-Verlag, 1995.
[14]Sontag, E. D., A ’universal’ construction of Artstein’s theorem on nonlinear stabilization, System & Control Letters, 1989, 13: 117–123. · Zbl 0684.93063 · doi:10.1016/0167-6911(89)90028-5
[15]Willems, J. L., Stability Theory of Dynamical Systems, London: Nelson, 1970.
[16]Deng, H., Krstic, M., Williams, R., Stochastic nonlinear stabilization-Part I: A backstepping design, Systems and Control Letters, 1997, 32: 143–150. · Zbl 0902.93049 · doi:10.1016/S0167-6911(97)00068-6
[17]Deng, H., Krstic, M., Williams, R., Stochastic nonlinear stabilization-Part II: A backstepping design, Systems and Control Letters, 1997, 32: 151–159. · Zbl 0902.93050 · doi:10.1016/S0167-6911(97)00067-4
[18]Tilbury, D. M., Sordalen, O. J., Bushnell, L. G., et al., A multi-steering trailer system: conversion into chained form using dynamic feedback, in Proceedings of the IFAC Symposium on Robot Control, Capri, Italy, 1994, 159–164; also IEEE Transactions on Robotics and Automation, 1995, 11(6): 807–818.
[19]Tilbury, D. M., Sastry, S. S., The multi-steering n-trailer system: a case study of Goursat normal forms and prolongations, in Symposium on Nonlinear Control System Design, Lake Tahoe, CA, 1995, 555–560.
[20]Tilbury, D. M., Laumond, J. P., Murray, R. et al., Steering car-like systems with trailers using sinusoids, in Proceedings of the IEEE International Conference on Robotics and Automation, 1992, Vol. 2, 1993–1998. · doi:10.1109/ROBOT.1992.219988
[21]Murray, R. M., Nilpotent bases for a class of non-integrable distributions with applications to trajectory generation for nonholonomic systems, Mathematics of Control, Signals, and Systems, 1995, 7(1): 58–75. · Zbl 0825.93319 · doi:10.1007/BF01211485
[22]Jiang, Z. P., Robust exponential regulation of nonholonomic systems with uncertainties, Automatic, 2000, 36: 189–209. · Zbl 0952.93057 · doi:10.1016/S0005-1098(99)00115-6
[23]Arnold, V. I., Geometrical Methods in the Theory of Ordinary Differential Equations, 2nd ed., Berlin: Springer-Verlag, 1987.
[24]Liu, Y. G., Zhang, J. F., Minimal-order observer and output-feedback stabilization control design of stochastic nonlinear systems, Science in China, Series F, 2004, 47(4): 527–544. · Zbl 1186.93065 · doi:10.1360/03yf0079
[25]Pan, Z. G., Liu, Y. G., Shi, S. J., Output feedback stabilization for stochastic nonlinear systems in observer canonical form with stable zero-dynamics, Science in China, Series F, 2001, 44(4): 292–308. · Zbl 1006.65045 · doi:10.1007/BF02878709
[26]Qksendal, B., Stochastic Differential Equations-An Introduction with Applications, New York: Springer-Verlag, 1995.
[27]Krstic, M., Kanellakopoulos, I., Kokotovic, P. V., Nonlinear and Adaptive Control Design, New York: Wiley, 1995.